1

Chemical Engineering 412

Introductory Nuclear Engineering

Lecture 10Nuclear Fission

Special Applications for Neutronics

• Neutrons are incoming particles• Energetics reveal likely behaviors

– Statistics! 1017-1023 particles in play– Describe behavior or high flux regions

(nuclear fuel/core)• Describe role of water in reactors• Background for neutron interactions

2

How to Decelerate a Neutron

Δ𝐸𝐸𝐸𝐸

=1 − 𝛼𝛼

2

𝑢𝑢 = ln𝐸𝐸𝑀𝑀𝐸𝐸

𝜉𝜉 = Δ𝑢𝑢 = 1 −𝐴𝐴 − 1 2

2𝐴𝐴ln𝐴𝐴 + 1𝐴𝐴 − 1

= 1 +𝛼𝛼

1 − 𝛼𝛼ln𝛼𝛼 ≅

2

𝐴𝐴 + 23

𝛼𝛼 =𝐴𝐴 − 1𝐴𝐴 + 1

2 collision parameter

Lethargy derivation: 𝐸𝐸𝑀𝑀 is an arbitrary 𝐸𝐸, usually the highest neutron energy in the system. As neutrons decelerate, 𝑢𝑢increases. (u = ln of energy change)

lim𝐴𝐴→1

𝜉𝜉 = 1

𝐸𝐸𝐸𝑚𝑚𝑚𝑚𝑚𝑚 = 𝛼𝛼𝐸𝐸 𝐸𝐸𝐸𝑚𝑚𝑚𝑚𝑚𝑚 = 𝐸𝐸

𝑚𝑚𝑚𝑚 𝑚𝑚𝑋𝑋

𝑚𝑚𝑚𝑚

𝑚𝑚𝑌𝑌

𝐸𝐸𝑚𝑚𝐸𝐸𝑋𝑋 = 0

𝐸𝐸𝐸

𝐸𝐸𝑌𝑌

𝜃𝜃𝑠𝑠𝜃𝜃𝑌𝑌

Neutron Energies

• Fission neutrons• Distribution of speeds• 2 MeV typical• Interested “slowing” neutrons• Collisions required to slow from energy 𝐸𝐸1 to 𝐸𝐸2 is given

by:

• Thermal neutrons:• equilibrated with the vibrating atomic nuclei at room

temperature (293 K)• Average energy of 0.025 eV (2200 m/s)• Maxwellian distribution of speeds• likely to lose OR GAIN energy from medium nuclei• Readily produce fissions in U235, U233, Pu239

𝑛𝑛 =1𝜉𝜉

ln𝐸𝐸1𝐸𝐸2

Collision parameters

Atom 𝑨𝑨 𝜶𝜶 𝝃𝝃 𝒏𝒏H 1 0.000 1.000 18.2H2O 1, 16 0.920 19.8D 2 0.111 0.725 25.1D2O 2, 16 0.509 35.8He 4 0.360 0.425 42.8Be 9 0.640 0.207 88.1B 11 0.694 0.171 106.3C 12 0.716 0.158 115.3O 16 0.779 0.120 151.7Na 23 0.840 0.084 215.4Fe 56 0.931 0.035 515.6238U 238 0.983 0.008 2171.6𝑛𝑛 values here assume a neutron slowing from 2 MeV to 0.025 eV

Capture and Absorption

• Decelerating Neutrons from fission energies (2-5 MeV) to thermal energies (0.025 eV)• Requires many collisions• Smaller Nuclides• Risk of “capture”

• Capture occurs in “resonance energy regions” (fuel)• Also could be absorbed by the “moderator” (water)• Can calculate probability of capture or absorption

• Resonance integral • Absorption cross-sections

Neutron Interactions

• Elastic scattering (n,n) – collision with no reaction and no change in total kinetic energies. Energy neutral.

• Inelastic scattering (n,n’) – collisions with energy absorption by nucleus. endoergic

• Radiative capture (n,γ) – Capture of neutron by nucleus followed by γ-ray emission. exoergic.

• Charged particle reactions (n,α) – Neutron reaction to form α particles or protons. endoergic and exoergic.

• Neutron producing reactions (n,xn) – Reactions with a net increase in neutrons. endoergic. (n,2n) important for 2H and 9Be.

• Fission (n, ) forms multiple products – Nucleus forms daughters. Generally exoergic.

Neutrons Eventually Are Captured

𝑛𝑛 + 𝑍𝑍𝐴𝐴𝑋𝑋 → 𝑍𝑍

𝐴𝐴+1𝑋𝑋 ∗ → 𝑍𝑍𝐴𝐴+1𝑋𝑋 + 𝛾𝛾

𝑛𝑛 + 90232𝑇𝑇𝑇 → 90

233𝑇𝑇𝑇 ∗ → 90233𝑇𝑇𝑇 + 𝛾𝛾

𝛽𝛽−→

22 𝑚𝑚91233𝑃𝑃𝑃𝑃

𝛽𝛽−→

27 𝑑𝑑92233𝑈𝑈

𝑛𝑛 + 510𝐵𝐵 → 5

11𝐵𝐵 ∗ → 37𝐿𝐿𝐿𝐿 + 𝛾𝛾 + 𝛼𝛼

𝑛𝑛 + 92238𝑈𝑈 → 92

239𝑈𝑈 ∗ → 92239𝑈𝑈 + 𝛾𝛾

𝛽𝛽−→

24 𝑚𝑚93239𝑁𝑁𝑁𝑁

𝛽𝛽−→

56 𝑇94239𝑃𝑃𝑢𝑢

Control rods

“Fertile” isotopes form “fissile” isotopes through neutron absorption

Fission Reactions

92235𝑈𝑈 is fissile (undergoes fission)

92238𝑈𝑈 is fertile (converts to a fissionable isotope)

𝑛𝑛 + 92235𝑈𝑈 →

92235𝑈𝑈 + 𝑛𝑛 𝑒𝑒𝑒𝑒𝑃𝑃𝑒𝑒𝑒𝑒𝐿𝐿𝑒𝑒 𝑒𝑒𝑒𝑒𝑃𝑃𝑒𝑒𝑒𝑒𝑒𝑒𝑠𝑠

92235𝑈𝑈 + 𝑛𝑛 + 𝛾𝛾 𝐿𝐿𝑛𝑛𝑒𝑒𝑒𝑒𝑃𝑃𝑒𝑒𝑒𝑒𝐿𝐿𝑒𝑒 𝑒𝑒𝑒𝑒𝑃𝑃𝑒𝑒𝑒𝑒𝑒𝑒𝑠𝑠

92236𝑈𝑈 + 𝛾𝛾 𝑠𝑠𝑃𝑃𝑑𝑑𝐿𝐿𝑃𝑃𝑒𝑒𝐿𝐿𝑟𝑟𝑒𝑒 𝑒𝑒𝑃𝑃𝑁𝑁𝑒𝑒𝑢𝑢𝑠𝑠𝑒𝑒𝑌𝑌𝐻𝐻 + 𝑌𝑌𝐿𝐿 + 𝑦𝑦1 + 𝑦𝑦2 + ⋯ 𝑓𝑓𝐿𝐿𝑒𝑒𝑒𝑒𝐿𝐿𝑓𝑓𝑛𝑛

Possible outcomes of 92235𝑈𝑈 reaction with neutron

Fission (logarithmic) Timeline

Scission (10-20 s)

Prompt Neutrons (10-17 s)

Gamma emission (10-14 s)

Return to background energy (10-12 s)

Absorption

Delayed Neutrons (1-100 s)

Fission reaction• Liquid drop model concept

– The already deformed nucleus deforms more significantly upon absorption such that the long dimension is long compared to residual strong force but short compared to Coulombic force, leading to net repulsion.

– The nucleus scissions, leaving two very highly charged (net charge +20 or so), highly energetic nuclei with far more neutrons than are stable.

– 0-8 prompt neutrons “boil off” the fission product nuclei (average number is �𝝂𝝂𝒑𝒑) within about 10-17 s

– Fission fragments decay from highly excited states to ground states typically via gamma emission within 10-14 s

– Still highly charged and very energetic fragments move through medium, neutralizing charge and reducing kinetic energy within 10-12 s

– A few (<1%) delayed neutrons are emitted in slow reactions such as neutron emission as quasi-stable fission products slowly decay.

https://www.youtube.com/watch?v=rA6pL4wUxXI

Emitted/Recoverable Energy

Critical Fission Energies

The energy release of a slow neutron added to a nucleus equals the binding energy of the last neutron added to the A+1 nucleus of the same isotope. If this

energy exceeds the critical energy for nuclear fission, then any neutron absorption will cause fission, and the nucleus is said to be fissile. All fissile nuclei have odd A.

A-1 Fissionable

A-1 Fissionable

A-1 Fissionable

A-1 Not Fissionable

A-1 Not Fissionable

Fission Product Distribution

Product Distribution at High Energy

Delayed Neutrons

• A small fraction (<1%) of total neutron production occur seconds or minutes after scission, represented by 𝜷𝜷 below. These delayed neutrons are essential to reactor control.

• Fast neutron emission alone is far too rapid to allow control.

Delayed Neutron Data

Prompt Neutron Energy

( )

( )bcbTEdEE

dEEEE

cEbEa

TEE

TET

EE

E

WW

W

W

Ww

w

w

+=+==

−=

+−

=

∫∫

∞

∞

642

3

)(

)(

sinhexp4sinhexp

0

0

2

χ

χ

πχ

Nuclide Neutron Energy

Ew Tw a b(Tw )

c(4Ew/Tw

2)�𝐸𝐸

233U thermal 0.3870 1.108 0.6077 1.1080 1.2608 2.05235U thermal 0.4340 1.035 0.5535 1.0347 1.6214 1.99239Pu thermal 0.4130 1.159 0.5710 1.1593 1.2292 2.15232Th 2 MeV 0.4305 0.971 0.5601 0.9711 1.8262 1.89238U 2 MeV 0.4159 1.027 0.5759 1.0269 1.5776 1.96252Cf spontaneous 0.359 1.175 0.6400 1.1750 1.0401 2.12235U (Lamarsh) 0.453 0.9653 2.29 1.98

( )Ww

WW

TETE

π/exp −

Neutron Energy Spectrum

Fission Energies pdf – log scale

Decay Heat

Fusion Basics• Coulombic repulsion

– (about 5-300 keV)– overcome by particle momentum

• In an accelerator (one particle at a time)• In a thermonuclear plasma – all particle energies exceed the Coulombic threshold.

• High temperatures– 10-300 MK – Sun’s core is 15 MK, minimum reactor temperature about 30 MK– Gases are fully ionized and form a plasma, – High pressures

• (400 billion bar at Sun’s core) 100,000 bar in plasma with particle density equal to atmospheric density• actual reactors operate up to 1 bar).

• Plasmas must be confined. Three approaches include– Gravitational Confinement (stars)– Magnetic Confinement (Tokamak and similar reactors)– Inertial Confinement (NIF and similar laser/photon systems)

• Deuterium in earth’s water (0.015% of H) could provide 50 billon years of energy at current consumption rates.

Primary Fusion Reactions

12𝐷𝐷 + 1

2𝐷𝐷 → 13𝑇𝑇 + 1

1𝐻𝐻 𝑄𝑄 = 4.03 𝑀𝑀𝑒𝑒𝑀𝑀

12𝐷𝐷 + 1

2𝐷𝐷 → 23𝐻𝐻𝑒𝑒 + 0

1𝑛𝑛 𝑄𝑄 = 3.27 𝑀𝑀𝑒𝑒𝑀𝑀

12𝐷𝐷 + 1

3𝑇𝑇 → 24𝐻𝐻𝑒𝑒 + 0

1𝑛𝑛 𝑄𝑄 = 17.59 𝑀𝑀𝑒𝑒𝑀𝑀

12𝐷𝐷 + 2

3𝐻𝐻𝑒𝑒 → 24𝐻𝐻𝑒𝑒 + 1

1𝐻𝐻 𝑄𝑄 = 18.59 𝑀𝑀𝑒𝑒𝑀𝑀

13𝑇𝑇 + 1

3𝑇𝑇 → 24𝐻𝐻𝑒𝑒 + 201𝑛𝑛 𝑄𝑄 = 11.33 𝑀𝑀𝑒𝑒𝑀𝑀

11𝐻𝐻 + 3

6𝐿𝐿𝐿𝐿 → 24𝐻𝐻𝑒𝑒 + 2

3𝐻𝐻𝑒𝑒 𝑄𝑄 = 4.02 𝑀𝑀𝑒𝑒𝑀𝑀

11𝐻𝐻 + 5

11𝐵𝐵 → 324𝐻𝐻𝑒𝑒 𝑄𝑄 = 8.08 𝑀𝑀𝑒𝑒𝑀𝑀

Of the many possible fusion reactions, the most practical for steady-state, land-based

power reactors involve reactants with singly positive

charges, which minimizes the Coulombic threshold

energy that must be overcome.

01𝑛𝑛 + 3

6𝐿𝐿𝐿𝐿 → 24𝐻𝐻𝑒𝑒 + 1

3𝑇𝑇 𝑄𝑄 = 4.78 𝑀𝑀𝑒𝑒𝑀𝑀

01𝑛𝑛 + 3

7𝐿𝐿𝐿𝐿 → 24𝐻𝐻𝑒𝑒 + 1

3𝑇𝑇 + 01𝑛𝑛𝑄𝑄 = −2.47 𝑀𝑀𝑒𝑒𝑀𝑀

Free neutron fusion with hydrogen provides the

deuterium and deuterium reactions and lithium

provide the tritium

Reaction Rate vs. Temperature

−𝑠𝑠 = 𝑓𝑓 = 𝑛𝑛1𝑛𝑛2 𝜎𝜎𝑟𝑟

• Volumetric reaction rate• Concentrations (n1, n2)• relative velocities (v)• effective size, or cross

section, (σ)• Real systems involve

ranges of velocities and cross section depends on energy/velocity, so an averages are used

• The cross section includes• Actual physical cross section• Reaction probability• One largely empirical parameter.

• Power production then is the rate times the energy of the reaction

𝑃𝑃𝑓𝑓𝑓𝑓𝑠𝑠 = 𝑛𝑛1𝑛𝑛2 𝜎𝜎𝑟𝑟 𝑄𝑄𝑓𝑓𝑓𝑓𝑠𝑠𝑚𝑚𝑖𝑖𝑚𝑚

Fusion Power Losses

• Plasma particles that interact with reactor walls are quenched to room temperature (and entrain wall material, typically Fe, into the plasma, which cools it).

• Plasmas are hydrodynamicallyunstable, generate turbulence, which leads to wall interactions.

• The primary radiative loss from a plasma is through free-free radiation or brehmsstralung – the emission of broadband radiation when free, charged particles (electrons and protons/deuterons/tritons) decelerate one another but remain free charges after the interaction.

Energy Losses

𝑃𝑃𝑙𝑙𝑖𝑖𝑠𝑠𝑠𝑠,𝑏𝑏𝑏𝑏𝑏𝑏𝑚𝑚 = 1.42𝑥𝑥10−34𝑍𝑍2𝑛𝑛𝑚𝑚𝑛𝑛𝑏𝑏 𝑇𝑇

Energy loss mechanisms include bremsstrahlung (primary), heating new fuel, synchrotron radiation at high temperatures, etc. The bremsstrahlung loss is

The temperature at which the losses equal the energy generated is the critical ignition temperature, or 𝑇𝑇𝑐𝑐, which for D-T and D-D is about 30 MK and 600 MK, respectively (see right).

𝑇𝑇𝑐𝑐 = 30 MK

𝑇𝑇𝑐𝑐 = 600 MK

Power Production and Loss

Gain Factor

To gain factor, 𝑄𝑄, is the ratio of the fusion energy production rate (power) to the required external power to maintain the system

𝑄𝑄 ≡𝑃𝑃𝑓𝑓𝑓𝑓𝑠𝑠𝑚𝑚𝑖𝑖𝑚𝑚𝑃𝑃ℎ𝑏𝑏𝑚𝑚𝑒𝑒

=1

𝜂𝜂ℎ𝑏𝑏𝑚𝑚𝑒𝑒𝑓𝑓𝑏𝑏𝑏𝑏𝑐𝑐𝑚𝑚𝑏𝑏𝑐𝑐𝜂𝜂𝑏𝑏𝑙𝑙𝑏𝑏𝑐𝑐 1 − 𝑓𝑓𝑐𝑐

where 𝜂𝜂ℎ𝑏𝑏𝑚𝑚𝑒𝑒 is the efficiency of electricity conversion to power to maintain the reactor (≈ 0.7), 𝑓𝑓𝑐𝑐 is the fraction of heat carried

away by the charged particles (≈ 1/5 for D-T), 𝜂𝜂𝑏𝑏𝑙𝑙𝑏𝑏𝑐𝑐 is the electrical power production efficiency from the reactor (≈ 0.35), 𝑓𝑓𝑏𝑏𝑏𝑏𝑐𝑐𝑚𝑚𝑏𝑏𝑐𝑐 is the fraction of power used to heat reactants (≈ 0.25),

run pumps, etc. ⇒ 𝑄𝑄 ≈ 20

Magnetic Confinement• Since plasmas consist of charged particles in motion,

they generate and are subject to magnetic forces. • A plasma with an average thermal energy of 10 KeV

(enough to overcome Coulombic repulsion) and a particle density equal to that in the atmosphere (1019

particles/cm3) requires about 100 kbar pressure. • A successful magnetic confinement reactor requires heat

extraction and subsequent power generation.• Deuterium-tritium reaction has most easily overcome

Coulombic interaction (single charge, heavy nuclide) requiring lowest plasma temperature

• Tritium generated by neutron-lithium reactions

12𝐷𝐷 + 1

3𝑇𝑇 → 24𝐻𝐻𝑒𝑒(3.54 𝑀𝑀𝑒𝑒𝑀𝑀) + 0

1𝑛𝑛(14.05 𝑀𝑀𝑒𝑒𝑀𝑀)

Magnetic Confinement

• By far the most successful – (though still unsuccessful) – Russian-developed Tokomak

reactor. • Shown is:

– the toroidal field and the coils (blue) that produce it,

– the plasma current (red) and the poloidal field produced by it,

– and the resulting twisted field when these are overlaid.

• MIT, Wisconsin have strong Tokomak programs

Tokamak Reactors

ITER

Gravitational Confinement

• Stars– The center of a star reaches temperatures of about 15

MK and pressures of 4 Gbar. • Such conditions are difficult to recreate on earth.• The average heat release of our sun is lower

than the average metabolic heat release of your body– Hot because it is so large, not because it is highly

reactive – average hydrogen nuclei lifetime in the sun is many

billion years• Highly heterogeneous, with nearly all fusion

reactions occurring in a relatively small core.

Inertial Confinement

• Inertial confinement uses inertial energy from rapid heating and implosion of reactants.

• Heating and implosion are typically driven by lasers (National Ignition Facility) or fissile explosions (weapons).

• Lawrence Livermore National Lab’s National Ignition Facility is a center of IC research and involves by far the world’s largest laser system.

• Lasers focus on a small cavity called a hohlraum(German for hollow room) that contains an even smaller (2 mm) spherical target that implodes under the force of the x-ray and laser photon energy.

NIF